2005

Abstract: It is known that the observability of a system crucially depends on the choice of the observable. Locally, such a feature directly results from the couplings between the dynamical variables (globally, it will also depend on symmetry). Using a feedback circuit description, it is shown how the location of the nonlinearity can affect the observability of a system. A graphical interpretation is introduced to determine - without any computation - whether a variable provides full observability of the system or not. Up to a certain degree of accuracy, this graphical interpretation allows to rank the variables from the best to the worst. In addition to that, it is shown that provided that the system is observable, it can be rewritten under the form of a jerk system. The Rössler system and nine simple Sprott systems, having two fixed points, are here investigated.

Abstract: Strange attractors can exhibit bifurcations just as periodic orbits in these attractors can exhibit bifurcations. We describe two classes of large-scale bifurcations that strange attractors can undergo. For each we provide a mechanism. These bifurcations are illustrated in a simple class of three-dimensional dynamical systems that contains the Lorenz system.

Abstract: The present paper extends some results recently developed for the analysis of observability in nonlinear dynamical systems. The aim of the paper is to address the problem of embedding an attractor using more than one observable. A multivariate nonlinear
observability matrix is proposed which includes the monovariable nonlinear and the linear observability matrices as particular cases.
Using the developed framework and a number of worked examples, it is shown that the choice of embedding coordinates is critical. Moreover, in some cases, to reconstruct the dynamics using more than one observable is clearly worse than to reconstruct using a scalar measurement. Finally, using the developed framework it is shown that
increasing the embedding dimension, observability problems diminish and can be even eliminated. This seems to be a physically meaningful interpretation of Takens embedding theorem.

Abstract: Spatio-temporal dynamics of a homogeneously broadened single-mode laser with large Fresnel number is investigated above the second laser threshold. The system is decribed by the two-level Maxwell-Bloch equations. A simple quasi-periodic regime is observed when the cavity is tuned below resonance, and very uncommon dynamics is obtained when the cavity is tuned above resonance. In the latter case, the laser intensity presents ``plateaux’’ of nearly constant values which are interrupted by bursts of large amplitude oscillations. The underlying dynamics is described in terms of heteroclinic connections between unstable periodic orbits associated with the constant intensities. A very surprising characteristic of this dynamics is that the periodic orbits are always visited in a given order which is related to the sequence of wave vectors selected by the laser. This new type of behavior presents many characteristics of intermittency.

Abstract: In the analysis of a scalar time series, which lies on an
-dimensional object, a great number of techniques will start by embedding such a time series in a -dimensional space, with . Therefore there is a coordinate transformation from the original phase space to the embedded one. The embedding space depends on the observable . In theory, the main results reached at are valid regardless of . In a number of practical situations, however, the choice of the observable does influence our
ability to extract dynamical information from the embedded attractor. This may arise in problems in nonlinear dynamics such as model building, control and synchronization. To some degree, ease of success will depend on the choice of the observable simply because it is related to the observability of the dynamics. In this paper the observability matrix for nonlinear systems, which uses Lie derivatives, is revisited. It is shown that such a matrix can be interpreted as the Jacobian matrix of - the map between the original phase space and the differential embedding induced by the observable - thus establishing a link between observability and embedding theory.

Abstract: When continuous systems are discretized, their solutions depend on the time step chosen a priori. Such solutions are not necessarily spurious in the sense that they can still correspond to a solution of the differential equations but with a displacement in the parameter space. Consequently, it is of great interest to obtain discrete equations which are robust even when the discretization time step is large. In this paper, different discretizations of the Lorenz system are discussed versus the values of the discretization time step. It is shown that the sets of difference equations proposed are more robust versus increases of the time step than conventional discretizations built with standard schemes such as the forward Euler, backward Euler or centered finite difference schemes. The non-standard schemes used here are Mickens’ scheme and Monaco and Normand-Cyrot’s scheme.